We seek solutions to the third order AL equation of motion that evolve
into the ``natural'' ones when the driving force is turned off. In
other words, radiation reaction must, by hypothesis, only damp the
system and not drive it. Clearly even this requirement makes no sense
when time reversal symmetry is considered. Once we fall into the trap
of choosing retarded interaction only, we are sunk and anything we
do to fix it will be a band-aid.

Let us introduce an ``integrating factor'' into the equations of motion.
If we assume (quite generally) that

(19.21)

where
is to be determined, then the equations of motion simplify
to

(19.22)

We can formally integrate this second equation, obtaining

(19.23)

The constant of integration is determined by our requirement that no runaway solutions exist! Note well that it is a constraint that
lives in the future of the particle. In order to use this to find
, we must know the force
for some time (of order
) in the future! After this, the integrand is ``cut off'' by the
decaying exponential.

This suggests that we can extend the integral to without
difficulty. In the limit , we recover Newton's law, as we
should. To see this, let

(19.24)

so that

(19.25)

The force is assumed to be slowly varying with respect to (or
none of this makes sense, just as was the case above) so that a Taylor
series expansion converges:

(19.26)

which, upon substitution and integration over , yields

(19.27)

In the limit only the lowest order term survives. This is
Newton's law without radiation reaction. The higher order terms are
successive radiative corrections and matter only to the extent that the force varies in time. Note that this force obeys a ``Lenz's Law'' sort of
behavior; when the applied force is changed (say, increased) there is an
additional ``force'' in the direction of the change that acts on the
particle. A particle moving in a circle has a force that changes direction but not magnitude. This change is (think about it) tangent to the
motion and in the opposite direction. It acts to slow the charged
particle down. Hmmmmmm.

There are two extremely annoying aspects to this otherwise noble
solution. First, as we have repeatedly noted, it requires a knowledge
of
in the future of the particle to obtain its
acceleration now. Truthfully, this isn't really a problem -
obviously this is absolutely equivalent to saying that
can be
expanded in a Taylor series (is an analytic function). Second, (and
even worse) it responds to a force that is completely in its
future with an acceleration now. It ``knows'' that a force is
going to act on it before that force gets there.

Figure:,
and
on a timescale of
. Note that the particle ``preaccelerates'' before ``the
force gets there'', whatever that means.

Mind you, not long before the force gets there. About
seconds before (for reasonable forces). Classically this is very bad,
but quantum theory fuzzes physics over a much larger time scale. This
is viewed by many physicists as an excuse for not working out a
consistently causal classical theory. You can make up your own mind
about that, but note well that even if the integrodifferential equation
had involved past values of the force you should have been equally bothered - either one makes Newton's law nonlocal in time!

Note well that we've already seen (nonlocal) integrodifferential
equations in time in a somewhat similar context! Remember our
derivation of of dispersion relations, in particular Kramers-Kronig? We
had a kernel there that effectively sampled times in the future or past
of a system's motion. This worked because we could integrate over frequencies with a constraint of analyticity - our fields were
presumed fourier decomposable. Fourier transforms are, of course,
infinitely continuously differentiable as long as we avoid sharp
changes like (pure) heaviside function forces or field changes, and yes,
they explicity provide a knowledge of the quantities in the future and past of their current values.

I personally think that this is yet another aspect of the mistake
made by requiring that our description of electrodynamics always proceed
from the past into the future with a retarded interaction. As we have
seen, this is silly - one could equally well use only advanced
interactions or a mix of the two and the solutions obtained for a given
boundary value problem will be identical, where the ``boundary''
is now a four-volume and hence requires future conditions to be
specified as well as the past conditions on a spatial
three-surface bounding the four-volume.